Deficit estimates for the Logarithmic Sobolev inequality

Abstract

Carlen showed that the Beckner-Hirschman entropic uncertainty principle, which is an improvement of Heisenberg's uncertainty principle, is equivalent to a quantitative logarithmic Sobolev inequality involving the entropy of the Fourier-Wiener transform. In this work, we identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. Inter-alia, we prove a new inequality relating the entropy of the Fourier-Wiener transform with the Fisher information and Wasserstein metric via the Monge-Ampère equation. Moreover, we investigate the dependence on the dimension of the stability constant and show that in certain configurations it can be taken independently and identify spaces in which the dimension-dependence cannot be removed. The techniques are based on optimal transport theory and Fourier analysis. Lastly, we discuss a probabilistic approach to stability estimates based on Cramér's convolution theorem.